Given that x and y are positive integers find x and y if log

Given that x and y are positive integers, find x and y if

log x + log y = x log y .

Solution

In the question we are given that log x + log y = x log y, where x and y are positive integers.

We use the results for logarithms here, which are log x + log y = log xy and x*log y = log y ^x.

Now here log x + log y = x log y

=> log xy = log y^x

Taking antilogs on both the sides: xy = y^x

Now as x and y are positive integers, this is possible only in the case where x=1 and y =2 . For no other set of positive integers does the relation x*y = y^x hold.

Therefore x = 1 and y =2.